Your mileage may vary, but I found that even though there were no outward signs of it, the timing of what remained of my hormonal cycle stayed roughly the same. So, if you are anything like me, the depression should lift around the same time it would if you were still having shark weeks.
As far as the symptoms of menopause go, I believe the fact that we are replacing our estrogen with testosterone keeps us from having hot flashes, irritability, and whatnot. It's the lack of significant quantities of EITHER sex hormone that causes menopause, I think. I certainly haven't had any hot flashes, or seen any other transguys speak of experiencing them.